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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 9, SEPTEMBER 2005
A Cramér–Rao-Type Estimation Lower Bound for
Systems With Measurement Faults
Ilia Rapoport and Yaakov Oshman, Senior Member, IEEE
Abstract—A Cramér–Rao-type lower bound is presented for
systems with measurements prone to discretely-distributed faults,
which are a class of hybrid systems. Lower bounds for both the
state and the Markovian interruption variables (fault indicators)
of the system are derived, using the recently presented sequential
version of the Cramér–Rao lower bound (CRLB) for general
nonlinear systems. Because of the hybrid nature of the systems
addressed, the CRLB cannot be directly applied due to violation
of its associated regularity conditions. To facilitate the calculation
of the lower bound, the hybrid system is first approximated by a
system in which the discrete distribution of the fault indicators is
replaced by an approximating continuous one. The lower bound is
then obtained via a limiting process applied to the approximating
system. The results presented herein facilitate a relatively simple
calculation of a nontrivial lower bound for the state vector of
systems with fault-prone measurements. The CRLB-type lower
bound for the interruption process variables turns out to be trivially zero, however, a nontrivial, non-CRLB-type bound for these
variables has been recently presented elsewhere by the authors.
The utility and applicability of the proposed lower bound are
demonstrated via a numerical example involving a simple global
positioning system (GPS)-aided navigation system, where the GPS
measurements are fault-prone due to their sensitivity to multipath
errors.
Index Terms—Estimation error lower bound, fault detection and
isolation, hybrid systems.
I. INTRODUCTION
M
ODERN multisensor applications, such as navigation
and target tracking systems, require the fusion of data
acquired by a large number of different sensors. In many situations these sensors might be subjected to faults, either due
to internal malfunctions, or because of external interferences.
These sensor faults are usually manifested as a sudden addition of noise (white or colored) to the sensor measurements,
or even interruption of the output signal. Thus, global positioning system (GPS) jamming and spoofing signals take, in
general, the form of colored noises and appear suddenly whenever they are activated [1]. Magnetometer faults, caused by magnetic fields generated by spacecraft electronics and electromagnetic torquing coils, usually take the form of biases [2, p. 251]
and appear whenever the corresponding current starts. Rate gyro
faults, caused by input accelerations if the gyro gimbals are not
perfectly balanced, are also usually modeled as biases [2, p.
198] and appear whenever the spacecraft accelerates. In view of
Manuscript received September 16, 2003; revised November 3, 2004 and May
1, 2005. Recommended by Associate Editor H. Wang.
The authors are with the Department of Aerospace Engineering of the
Technion—Israel Institute of Technology, Haifa 32000, Israel (e-mail:
[email protected]).
Digital Object Identifier 10.1109/TAC.2005.854579
present day systems’ high accuracy requirements, the problem
of fault tolerant filtering in multisensor systems is of major importance.
Characterized by sudden structural changes, fault-prone
system behavior is usually modeled and analyzed using the
framework of hybrid systems [3, p. 177]. The total state of these
systems comprises two kinds of parameters: the continuously
distributed parameters, usually referred to as the system states,
and a Markovian switching parameter, which takes values in a
finite set and is referred to, in general, as the system mode. Considering fault-prone systems, one of the switching parameter
values corresponds to the nominal system operation, whereas
the others represent various fault conditions [1], [4]. In systems
with independent fault sources and fault-free dynamics, such
as GPS-aided inertial navigation systems, the aforementioned
model can be simplified: The faults caused by different sources
can be modeled as separate Markovian Bernoulli random processes, where “1” stands for a fault situation and “0” stands for
no fault situation. Since the state vector is free of faults, these
fault indicators affect only the system measurements.
Because of its importance, the problem of stability and
control of hybrid systems has drawn considerable research
efforts over the past several decades. Depending on the particular engineering problem, a varying level of certainty in the
hybrid models was assumed in various works. Thus, in [3] and
[5]–[7] the stability and control of hybrid systems was investigated under the assumption that both the state and the mode
variables are known. In other works (see, e.g., [8] and [9]), a
partial-knowledge structure was adopted: only the state vector
was assumed to be known, whereas the system mode was not
directly observed and, consequently, had to be estimated.
In systems with fault-prone sensors, direct access to both the
state and the mode parameters is naturally unavailable, so that
the problem of their simultaneous estimation is of prime importance. It is well-known that the mean square optimal filtering
algorithm for hybrid systems, that provides the estimates of the
state vector and the switching parameters, requires infinite computation resources [10]. Therefore, a variety of suboptimal estimation techniques was proposed [11]–[16]. Since the estimates
of the state vector and the fault indicators are suboptimal, it is
of particular interest to obtain some measure of their efficiency.
The natural means for this purpose is the comparison to a lower
bound on the estimation error.
The most popular lower bound is the well-known
Cramér–Rao lower bound (CRLB). This bound is presented
in [17, p. 84] in the context of Bayesian estimation of static
random parameters. In that formulation, also known as the
Van Trees version of the CRLB, the underlying static random
0018-9286/$20.00 © 2005 IEEE
RAPOPORT AND OSHMAN: A CRAMÉR–RAO-TYPE ESTIMATION LOWER BOUND
system is assumed to satisfy the so-called CRLB regularity
conditions [17, p. 72], that is, absolute integrability of the
first two derivatives of all related probability density functions
(pdfs). Later, [18] and [19] provided a CRLB derivation under
less restrictive requirements, but even these weaker conditions
state, among others, that all related pdfs must be continuously
differentiable [19].
The first derivation of a sequential CRLB version applicable to discrete-time dynamic system filtering, the problem
addressed in this paper, was done in [20] and then extended
in [21]–[23]. Recently, the most general form of sequential
CRLB for discrete-time nonlinear systems was presented in
[24]. Together with the original static form of the CRLB, these
results served as a basis for a large number of applications
[25]–[29].
Unfortunately, albeit being a very useful tool for systems with
continuously distributed parameters, the CRLB cannot be directly calculated for both the state and the mode variables of
a hybrid system. The reason lies in the fact that the system of
interest must satisfy the CRLB regularity conditions. Clearly,
this is not the case in hybrid systems in general and in systems
with fault-prone measurements in particular, since the mode
variables, or fault indicators, are discrete Markovian sequences.
The CRLB can be applied directly only to the state vector (see,
e.g., [29]), which is continuously distributed and, in most cases,
satisfies the regularity conditions. However, the application of
the sequential form of the CRLB [24] is based on a special requirement regarding the structure of the system measurements,
which, as will be shown later, is not satisfied by a general class
of systems with fault-prone measurements. Another approach
is based on treating the discretely-distributed fault indicators
as nuisance parameters, known to the observer. Originally proposed in [30] for a general class of systems, this approach was
applied in [31] to target tracking using sensors with detection
probability smaller than one. Using the fact that the measurement interruption process is white the authors derived an approximation of the CRLB for the tracking errors. However, in
the case of general (nonwhite) Markov sequences such derivation becomes cumbersome. Moreover, this lower bound cannot
be evaluated in closed form, calling for the use of extensive
Monte-Carlo simulations.
A new approach for the calculation of a CRLB-type lower
bound for hybrid systems is presented in this paper. The approach is based on the approximation of the discrete distributions of the fault indicators by appropriate smooth distributions.
Satisfying the regularity conditions, the approximating distributions allow the calculation of the CRLB using its sequential
version for general discrete-time nonlinear systems [24]. The
CRLB-type lower bound for the original hybrid system is then
obtained via a limiting process applied to the approximating
system. Unlike the result of [30], the bound presented herein can
be evaluated in closed-form. Therefore, its application to complex systems entails only a modest computational load. Moreover, this result enables the practitioner to analytically evaluate
the effects of various system parameters on the attainable estimation performance. The utility of the proposed lower bound,
as well as the simplicity of its application, are demonstrated via
a numerical example involving a GPS-aided navigation system.
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The remainder of this paper is organized as follows. The
system model is defined and the problem is formulated in
Section II. The underlying idea behind the derivation of the
proposed lower bound is presented and justified in Section III.
The formal derivation procedure is then detailed in Section IV.
The main result of this paper, namely, the lower bound for
systems with fault-prone measurements, is presented and
discussed in Section V. A numerical example demonstrating
the application of the new lower bound to the assessment of
navigation accuracy in a simple GPS-aided navigation system
is presented in Section VI. Concluding remarks are offered
in the final Section. To enhance readability, some auxiliary
calculations and developments are deferred to Appendices.
II. PROBLEM FORMULATION
Consider the system
(1a)
(1b)
is a sequence of (known) deterministic inputs,
and
are white sequences of process and
measurement noise, respectively, with
and
is the random initial
state with
is the vector of
is a Bernoulli
mode variables, and every sequence
Markov chain with distribution
where
(2a)
(2b)
, are homogenous, i.e.,
It is assumed that the chains,
are constant in time.
the transition probabilities
and the chains
It is also assumed that
, are mutually independent. In addition, the observation matrix is assumed to satisfy the following
structural constraint:
(3)
The hybrid system defined previously is denoted by .
For notational simplicity the explicit time-dependence is suppressed in the sequel in all places where it is clear by context.
Also, the transition matrix is assumed to be nonsingular.
The model defined above can represent a wide class of
systems in the area of fault detection and isolation. In these
comprises two parts: The primary
systems, the state vector
part is associated with the system dynamics and the secondary
part is associated with the dynamics of the sensors when sensor
faults occur. These fault states can describe various kinds of
sensors’ faulty behavior, e.g., measurement biases, or additive
faulty measurement noises (white or colored). The interruption
play the role of fault indicators. An example of
variables
such a system is presented in Section VI.
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The following definitions will be used in the sequel. Let
(4)
be the augmented state vector of the system. Denote, also, the
accumulated history of the measurements as
(5)
based on the measurement history
Any estimator of a vector
is denoted as
. Let
(6)
and
(7)
In addition, define the operator
as
(8)
where
(9)
is the gradient operator. Finally, for presentation clarity, the notational convention of [17] is adopted, according to which lower
case and upper case letters are used to denote the pdf’s associated random variables and their realizations, respectively.
The goal of this work is to derive a lower bound for the estimation error covariance matrix of the augmented state vector
(4).
III. UNDERLYING IDEA
In this section, the idea underlying the derivation of the new
lower bound is proposed and justified. The derivation is based
on the sequential form of the CRLB [24], which is summarized
in the next theorem.
Theorem 3.1 (Tichavský, Muravchik, and Nehorai,
1998): Let
denote a Markovian state process.
Let
denote the corresponding measurement process,
assumed to be of the form
(13c)
Remark 3.1: It follows from the derivation presented in [24]
are blocks of the Fisher information matrix
that the matrices
corresponding to the joint distribution of the history of all the
and all the measurements
. Moreover,
states
is the
block of the inverse of this global Fisher
information matrix.
Remark 3.2: If the CRLB regularity conditions are satisfied,
the resulting Fisher information matrix is nonsingular [18]. It
follows, therefore, that the Fisher information submatrix
is also nonsingular.
Straightforward use of the sequential CRLB, presented
above, to compute an estimation bound for both the state
and the fault of the system , defined in Section II, is impossible. This follows from the discrete nature of the distribution
of the fault vector, which violates the requirement that the joint
pdf of the state vector and the measurement vector be twice
differentiable. Moreover, since in the system addressed in this
paper the measurements are related to the state through
(14)
the sequential CRLB cannot be applied to the primary state, ,
is white, because the structural requirealone, unless
ment (10) is not satisfied.
To circumvent the aforementioned problem, it is proposed
herein to adopt the following two-stage derivation procedure. In
the first stage, the original hybrid system is approximated by
a continuous system which is identical to except that the discrete distribution of is replaced by a continuous one. Specifically, the Bernoulli Markov chain
is replaced by a
continuously distributed Markov process
. The transition and initial pdfs of this new process are defined as
(15a)
(10)
where
noise. Then
is a white sequence independent of the process
(11)
where , termed Fisher information submatrix in [24], is computed sequentially according to the recursion
(12a)
(12b)
with
(13a)
(13b)
(15b)
where is some positive real parameter,
is the probability
has the following propdefined in (2b) and the function
erties:
1)
is twice continuously differentiable;
, namely,
2)
the first two derivatives of
and
, are bounded;
3)
and
;
4)
and
;
is monotonic.
5)
Notice that each pdf defined in (15) has the form of a two-peak
function. The steepness of the peaks is determined by the parameter . Intuitively it is clear that the smaller the value of ,
RAPOPORT AND OSHMAN: A CRAMÉR–RAO-TYPE ESTIMATION LOWER BOUND
the better the approximation of the discrete distribution (2) by
the continuous distribution (15). A thorough discussion of the
properties of the distribution (15) can be found in [32].
the continuous system resulting from
Now, denote by
the discrete distribution (2) by the continuous
replacing in
. As shown in Appendix B,
distribution (15) for some
the resulting joint pdf of the measurements and the estimated
variables satisfies the regularity conditions even in their most
restrictive form [17, p. 72]. Thus, the recursive CRLB corcan be computed for any
. In the
responding to
second stage of the derivation, a limiting process is applied to
. As the next two theorems show, this
this CRLB as
two-stage procedure leads to a lower bound for the original
approximates
system , since: 1) the continuous system
in the sense that the estimation error
the hybrid system
covariance matrices resulting from the application of any
admissible estimator to both systems can be made arbitrarily
, and 2) the CRLB obtained for
close by letting
provides a lower bound for the original hybrid system via a
limiting procedure, as
.
To present the next theorems let be any estimator satisfying
and
be the estimathe CRLB requirements. Let
tion error covariance matrices resulting from the application of
to
and to
using measurements up to and including
. In addition, let
be
time , respectively, where
the Fisher information submatrix obtained by applying the se.
quential CRLB to
Theorem 3.2: The system
can be approximated by systems with continuously distributed interruption variables [as in
(15)] to an arbitrary degree of accuracy, in the sense that
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defined by (20) are continuous functions of
is assumed that
’s. To this end it
(21a)
(21b)
meaning that the estimation errors have finite secondand fourth-order moments. Now, the conditional pdf
is Gaussian. Since ’s
appear linearly in (3), the mean and the covariance matrix of
this distribution are continuous functions of ’s. On the other
hand, by Lemma A.1, the integral defined in (20) and satisfying
conditions (21) is a continuous function of the mean and the
covariance matrix. Therefore, this integral is a continuous
function of ’s.
To demonstrate (16), it is sufficient to show that
(22)
First, notice that due to the Markov property
(16)
Proof: The proof makes use of some results that are presented in Appendix A (and numbered accordingly). Let
(17)
(23)
(18)
Then, the smoothing property of conditional expectation yields
where each of the conditional pdfs is given by (15a). It will be
shown now that
(19a)
(19b)
where the inner conditional expectations in both (19a) and (19b)
are identical matrix functions of some sets of values of ’s and
’s, respectively. It will be shown now that the
element
, is a continuous function
of these expectations, denoted by
of the conditioning variables.
First, it can be seen that every element of the conditional expectations can be expressed as a second-order polynomial of the
conditioning variables with coefficients of the form
(20)
where the function
is determined by the particular
estimator. Therefore, it is sufficient to show that the integrals
(24)
Notice that
(25)
As previously mentioned,
is a continuous function of
. In addition, by Lemma A.2, being an element of a pos-
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itive–semidefinite matrix,
is absolutely integrable with
. Therefore,
respect to the conditional distribution of
by Lemma A.3
and the elements of
are mutually independent, the
Since
augmented state vector has the marginal density
(32)
and the following transitional density:
(26)
and (24) is obtained recalling that
(33)
(27)
and using the fact that
Applying these arguments for each
are continuous functions of finally yields (22).
Theorem 3.3: The Fisher information submatrix
, obtained by applying the CRLB to the approximating continuous
, provides an estimation error lower bound for the
system
hybrid system via a limiting process, namely
The distribution of
is Gaussian with
(34)
Therefore
(35)
(28)
is, by definition, the Fisher informaProof: Since
tion submatrix computed for the approximating continuously
, then
distributed system for some
where
(29)
Therefore, Theorem 3.2 yields
(36)
(30)
and
(37)
The distribution of the measurements conditioned on
Gaussian, hence
IV. LOWER BOUND DERIVATION
This section presents a derivation of the new lower bound,
based on the underlying idea proposed in Section III. The new
lower bound itself, which constitutes the paper’s main result,
is presented in the next section. The derivation comprises two
steps: 1) application of the sequential CRLB to the approxifor some
, and 2) commating continuous system
. For the sake of brevity
putation of the CRLB limit as
only the main highlights of the derivation are presented. The interested reader is referred to [32] for a detailed derivation.
is initially asTo facilitate the derivation the matrix
sumed to be nonsingular. This assumption is relaxed in the sequel via a procedure proposed in [24].
is also
(38)
Using (35) and (38) in (13), the following expressions for the
matrices, defined in Theorem 3.1, are obtained
(39a)
(39b)
A. CRLB for
Analogously to (4), let
(39c)
(31)
In (39), the following terms are used.
RAPOPORT AND OSHMAN: A CRAMÉR–RAO-TYPE ESTIMATION LOWER BOUND
1)
is a diagonal matrix whose th diagonal entry
is
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in (47), the exTo avoid the need to explicitly invert
pression (47) can be rewritten in the following form, using the
well-known matrix inversion lemma [33, p. 19]:
(40)
is a diagonal matrix whose th diagonal entry
2)
is
(41)
(48)
3)
The matrix
is given by
(42)
4)
The
element of
is
(43)
5)
The th column of
B. Limiting Case
The limiting value of the CRLB for
is presented in
the following theorem.
, the inverse
Theorem 4.1: For every finite time instant
of the limiting Fisher information submatrix satisfies the following relation:
is
(49)
(44)
6)
is a diagonal matrix whose th diagonal entry is
where
is a positive–definite matrix satisfying
the following recursion:
(45)
(50a)
(50b)
It is shown in [32] that
(46a)
(46b)
(46c)
is the discretely-distributed fault indicator vector
In (50a)
of the original hybrid system.
Proof: The proof uses mathematical induction. First,
using (12b) and following the derivation procedure of (39c), it
can be shown that, for
Now, substituting (39) into (12a) yields the following propagation formula for the Fisher information submatrix:
(51)
In (51)
is a diagonal matrix, whose th diagonal entry is
(52)
that satisfies
(47)
Since in the approximating system
the CRLB regularity
conditions are satisfied, Remark 3.2 renders the global Fisher
information matrix nonsingular. Therefore, the Fisher informais also nonsingular.
tion submatrix
(53)
Therefore
(54)
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which implies both (49), for
, and (50b). Notice that the
follows from that of .
positive–definiteness of
Now, assuming that the Theorem is valid for the finite time
instant , it will be shown that it also holds for time instant
.
The induction recursion is proved in the following three steps.
Step
1)
Using continuity arguments, the induction assumption,
the matrix inversion lemma and the fact that, by (46a)
and (46b)
is also nonsingular. Using Schur’s formula for the inversion of
partitioned matrices [33, p. 18] yields
(61)
In addition, the expression
pends only on the first and second moments of
de. Therefore
(55)
(62)
yields
which yields
(63)
(56)
Step
2)
Applying the matrix inversion lemma to the inverse on
the right-hand side (RHS) of (48) and using (46b) and
(56) yields
Substituting (63) into (61) gives (58). The positive–definiteness
follows from (50a) where the second term on the RHS
of
is nonsingular.
Finally, the assumption regarding the regularity of
,
made at the beginning of the derivation, is relaxed in the following theorem.
Theorem 4.2: Theorem 4.1 applies also in systems where
is singular.
Proof: The proof is motivated by a procedure shown in
by
for some small positive
[24]. Replace
. This new matrix is nonsingular and, therefore, Theorem 4.1
applies. Sending to zero and using the continuity of the second
term on the RHS of (50a) yield a CRLB-type lower bound for a
, which is also given by (50a).
singular
(57)
V. MAIN RESULT
Step
3)
Finally, it is shown that
(58)
The main result of this paper is now stated in the following
theorem.
Theorem 5.1: A CRLB-type lower bound for the system defined in Section II is given by
(64a)
is calculated using (50a). Notice that all the
where
, are finite for
terms on the RHS of (48), except for
. Therefore, using (48)
(59)
with implied definitions of the finite matrices
, and . Since
(64b)
is computed using the recursion (50).
where
Proof: The Theorem follows immediately from Theorems
3.3 and 4.2.
A. Lower Bound Computational Algorithm
(60)
The lower bound presented in Theorem 5.1 can be easily evaluated using the following recursive algorithm.
1)
Initialize the matrix
using (50b).
update the a priori
2)
At each time step
probabilities
the matrix is nonsingular for sufficiently small values of .
Similarly, for sufficiently small values of the term
(65)
RAPOPORT AND OSHMAN: A CRAMÉR–RAO-TYPE ESTIMATION LOWER BOUND
3)
Update the value of
term
using (50a), where the
can be computed as
(66)
4)
Finally, the CRLB-type lower bound is given by (64).
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5.1. This bound is regarded in the present context as a MonteCarlo lower bound (MCLB).
Remark 5.1: Since the MCLB explicitly assumes that the
fault vector sequence is known to the observer, it does not take
into account the effect of the fault vector estimation error on the
state vector estimation accuracy.
The next proposition relates the lower bound presented in
Theorem 5.1 to the MCLB.
Proposition 5.1: The lower bound presented in Theorem 5.1
is less tight than the MCLB, i.e.,
(70)
B. Lower Bound Properties
The result given by (64a) and (50) presents a relatively
simple, nontrivial lower bound on the covariance of the estimation error of the state vector . Similarly to the covariance
recursion in the information form of the Kalman filter (KF),
this bound can be easily computed and used to evaluate the performance of estimators designed for systems with fault-prone
measurements.
On the other hand, the lower bound on the estimation error
covariance of the fault indicators, given by (64b), is trivial. An
intuitive reason for this result lies in the fact that the fault indica, being discretely distributed parameters, are charactertors
ized by sharp changes in the cumulative distribution function. It
has been reported previously in the literature [34] that the CRLB
tends to zero in systems with sharp intensity functions. A direct
conclusion from this observation is that in order to obtain a nontrivial lower bound on the discretely distributed fault indicators
one must examine lower bounds that are not based on the CRLB.
An example of such a lower bound has been recently presented
in [35].
Proof: It is first proved that
(71)
The proof is by mathematical induction. First, according to
(50b)
(72)
Assume now that inequality (71) holds for some . Then, (50a),
(68), and Lemma A.4 of Appendix A yield
(73)
so that (71) is established.
Now, it follows from (71) that
(74)
Finally, Lemma A.4 yields
C. Comparison to a Monte-Carlo Lower Bound
Yet another lower bound can be obtained for the system by
treating the fault vector sequence
as nuisance parameters that are known to the observer. Originally proposed in [30],
this approach is as follows. Assume that the fault vector sequence is known. Under this assumption, the addressed system
takes the form of an ordinary linear Gaussian system, yielding
the following CRLB:
(67)
where the Fisher information matrix is governed by the following recursion:
(68)
Taking mathematical expectation of both sides of (67) gives
(69)
The main drawback of this lower bound is the fact that it cannot
be evaluated in closed-form. In practice, extensive Monte-Carlo
simulations must be performed, rendering the computation of
the bound rather difficult, compared to the result of Theorem
(75)
Remark 5.2: The arguments presented here can be interpreted as an alternative derivation of the lower bound given
by Theorem 5.1. Moreover, this derivation is valid for a larger
class of systems than that of Theorem 5.1 because no specific
assumptions were made on the structure of the observation
, as was done in (3). However, this derivation
matrix
does not demonstrate the fact that the proposed lower bound is
a limiting case of the CRLB.
Corollary 5.1: The lower bound presented in Theorem 5.1
does not take into account the effect of the fault vector estimation errors on the state vector estimation accuracy.
Proof: The Corollary follows from Proposition 5.1 and
Remark 5.1.
VI. NUMERICAL EXAMPLE
A numerical proof-of-principle example is presented to
demonstrate the applicability and utility of the new lower
bound. The example involves a system comprising an inertial
navigation system (INS), a GPS receiver and an additional po,
sition source, e.g., a terrain following system. Let
and
denote the INS position error, velocity error and
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GPS receiver clock drift, respectively. The following simplified
dynamics is assumed:
TABLE I
NUMERICAL STUDY PARAMETERS
(76a)
(76b)
(76c)
where is the speed of light. The clock drift noise,
, is
assumed to be a zero-mean, Gaussian white noise with power
,
spectral density (PSD) equal to . The velocity noise,
is also assumed to be zero-mean, white and Gaussian with PSD
.
The measurements in this navigation system are taken at a
rate of 1 Hz and the continuous-time system (76) is discretized
accordingly. Assume that the GPS receiver tracks 4 satellites.
denote the GPS pseudorange error, defined as
Let
(77)
is the range from the receiver’s position, as computed
where
is the pseudorange to the
by the INS, to the th satellite, and
th satellite as measured by the GPS receiver. Each of the GPS
receiver channels may be subjected to a multipath measurement
error, so that the pseudorange error is given by the following
linear equation:
(78)
for
, where
is the unit direction vector to
is the pseudorange measurethe th satellite,
is a multipath parameter and
indiment white noise,
cates the presence of multipath error. For simplicity, the satellite
constellation is assumed to be constant throughout the scenario
with geometric dilution of precision (GDOP) equal to 2.13. The
multipath parameters are assumed to behave as a stationary,
zero-mean, Gaussian colored noise with autocorrelation
(79)
which corresponds to the following state–space model
(80)
where
satisfying
is a zero-mean, white process noise
(81)
In addition to the GPS measurements the navigation system
makes use of a terrain-following system, generating measurements modeled as:
TABLE II
TEST CASE PARAMETERS
than those of the GPS receiver. On the other hand, these measurements are assumed to be fault-free over the scenario duration. The numerical example parameters are summarized in
Table I.
The newly proposed lower bound and the MCLB are used to
assess the estimation performance of the following two filters.
1)
The first filter utilizes the fault-free terrain-following
measurements only. This filter serves as a baseline
filter with a minimal measurement configuration.
Notice, that this is the only minimal measurement
configuration: the other potential minimal configuration, namely, the configuration utilizing the fault-prone
GPS measurements only, renders the system described
above not completely observable. Since the system
with terrain-following measurements only is linear
and Gaussian, the baseline filter is a standard KF.
2)
The second filter is designed to examine the possible
estimation accuracy contribution of the fault-prone
GPS measurements, when used in addition to the baseline terrain-following measurements. Since the total
system is hybrid, it is proposed to use the interacting
multiple model (IMM) algorithm [14], which is known
to be a powerful tool in filtering of hybrid systems.
1000 Monte-Carlo runs are used to evaluate the performance
of the KF and 400 Monte-Carlo runs are used to evaluate the
performance of the IMM filter. Three test cases, whose numerical parameters are summarized in Table II, are studied. In all
, are assumed to be
three cases the initial fault probabilities,
zero. The position and velocity estimation performance metrics
used in this study are defined as the root-sum-square (RSS) of
the components of the position and velocity root mean square
(RMS) estimation errors, respectively, namely
(83)
(82)
where
is the measurement white noise. The
terrain-following measurements are assumed to be less accurate
where
, is the estimation error in the position
or velocity component at the th time step of the th Monteis the total number of Monte-Carlo runs.
Carlo run, and
RAPOPORT AND OSHMAN: A CRAMÉR–RAO-TYPE ESTIMATION LOWER BOUND
Fig. 1. Estimation errors achieved by IMM (bold solid line) and KF (thin solid
line) vs the CRLB-type lower bound (thin dashed line) and the MCLB (thin
dashed–dotted line) in the nominal case. (a) Position. (b) Velocity.
The results of the first case study are presented in Fig. 1, that
shows the position [Fig. 1(a)] and velocity [Fig. 1(b)] estimation
errors. Comparing the absolute errors of the filters one can see
that the IMM filter is about two times more accurate than the KF in
position estimation [see Fig. 1(a)] and about 1.5 more accurate in
velocity estimation [see Fig. 1(b)]. However, a direct comparison
to the CRLB-type lower bound reveals that the IMM is much
more efficient (in the statistical sense) than the KF: The position
estimation error is about 3 times closer to the lower bound than
the position estimation error of the KF. Considering the velocity
estimates one can see that the IMM velocity error almost reaches
the lower bound. This means that the IMM velocity estimates are
almost optimal in the minimum variance sense, and also that the
proposed lower bound is tight in this case.
The results of the second case are presented in Fig. 2. Due
to the high multipath probability, the position error of the IMM
filter and the corresponding CRLB-type lower bound increase
[see Fig. 2(a)]. The KF performance remains about the same as
in the previous case, since the KF estimates do not depend on
the GPS measurements. Again, whereas its position estimate is
better by only 70% than that of the KF, a direct comparison to
the lower bound reveals that the IMM is 3.6 times more efficient
than the KF, and its velocity estimate can be regarded as optimal
(rendering the corresponding lower bound tight).
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Fig. 2. Estimation errors achieved by IMM (bold solid line) and KF (thin
solid line) vs the CRLB-type lower bound (thin dashed line) and the MCLB
(thin dashed–dotted line) in the high multipath probability case. (a) Position.
(b) Velocity.
Examining the third case (see Fig. 3) one can notice that due
to the low process noise the estimation errors of both filters, as
well as the CRLB-type lower bounds, diminish. Now the velocity estimates of both filters can be considered as optimal [see
Fig. 3(b)]. As for the position errors, the KF performance becomes closer to that of IMM filter relatively to the previous two
cases: The IMM estimation error is only 1.2 times better than
that of the KF and only 1.6 times closer to the lower bound.
Figs. 1–3 also show the MCLB of (69). 100 Monte-Carlo runs
were used to evaluate this bound, which is associated with a
computational load a hundred times larger than that of the proposed CRLB-type lower bound. One can see, especially in the
position errors [Figs. 1(a), 2(a), and 3(a)], that the CRLB-type
lower bound is indeed less tight than the MCLB, as has been
predicted in Proposition 5.1.
VII. CONCLUSION
A CRLB-type lower bound for a class of systems with faultprone measurements is presented. Lower bounds for both the
state and the Markovian interruption variables (fault indicators)
of the system are derived, based on the recently presented sequential version of the CRLB for general nonlinear systems. The
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 9, SEPTEMBER 2005
APPENDIX A
AUXILIARY RESULTS
In this appendix, some auxiliary results are presented, that
are used in the derivation and discussion of the lower bound
(Sections IV and V). The proofs of the Lemmas are omitted for
brevity, and can be found in [32].
be a Gaussian pdf family with
Lemma A.1: Let
and denoting the mean and covariance matrix of each of its
be some particular values of
members, respectively. Let
, with
. Then, for a function
that satisfies
(A.1a)
(A.1b)
for all
that belong to some neighborhood of
following holds true:
, the
(A.2)
Lemma A.2: Let
be the pdf of a random vector . Let
be some positive–semidefinite matrix such that
(A.3)
Then, every element of
sense that
Fig. 3. Estimation errors achieved by IMM (bold solid line) and KF (thin solid
line) vs the CRLB-type lower bound (thin dashed line) and the MCLB (thin
dashed–dotted line) in the low process noise case. (a) Position. (b) Velocity.
derivation is based on an approximation of the discrete distribution of the interruption variables by a continuous one. The lower
bound is then obtained via a limiting process applied to the approximating system.
The results presented in this paper facilitate a relatively
simple calculation of a nontrivial CRLB-type lower bound for
the state vector of systems with fault-prone measurements.
Application of the CRLB-type lower bound to both the state
vector and the measurement interruption variables does render
the bound for the interruption process variables trivially zero,
and shows that it is unable to take into account the effect of
the fault vector estimation errors on the state vector estimation
accuracy. However, an alternative, non-CRLB-type nontrivial
lower bound for the interruption variables has been recently
presented elsewhere by the authors.
The utility and applicability of the proposed lower bound
are demonstrated via a numerical example involving a simple
navigation system aided by a fault-prone GPS receiver and terrain-following position measurements. It is shown that the new
lower bound serves as an efficient tool in the design of filters for
this fault-prone system, as it facilitates the assessment of candidate filters, designed for different measurement system configurations.
is absolutely integrable in the
(A.4)
Lemma A.3: Let
be a function, continuous at some
, that satisfies
point
(A.5)
Then
(A.6)
Lemma A.4: Let
be a positive–definite random matrix
with nonsingular mathematical expectation, and let be some
deterministic positive–semidefinite matrix of the same dimensions. Then
(A.7a)
(A.7b)
APPENDIX B
REGULARITY CONDITIONS
The regularity conditions for the system defined by (1),
(3), and (15) are that the first two derivatives of the joint pdf
with respect
’s and ’s are absolutely integrable, and that the first
to
RAPOPORT AND OSHMAN: A CRAMÉR–RAO-TYPE ESTIMATION LOWER BOUND
moments of the estimation errors are finite (see [17, p. 72]).
Notice that
(B.1)
Now, all the pdfs in (B.1) are either Gaussian or linear
combinations of Gaussian distributions with coefficients
’s. Recalling that the functions
depending on
’s are twice differentiable with bounded derivatives yields absolute integrability of the joint pdf
.
As for the requirement that the first moments of the estimation errors be finite, the discussion is restricted to those estimators that produce finite second moments of the estimation error.
Therefore, this requirement is naturally satisfied.
ACKNOWLEDGMENT
The authors would like to thank the anonymous reviewer for
constructive suggestions and comments.
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P <
Ilia Rapoport was born in Ufa, U.S.S.R., in January, 1971. He received the
B.Sc. (summa cum laude) and M.Sc. degrees in aerospace engineering from the
Technion—Israel Institute of Technology, Haifa, in 1994 and 2000, respectively.
He is currently working toward the Ph.D. degree in the area of estimation theory
at the Technion’s Department of Aerospace Engineering, under the supervision
of Prof. Y. Oshman.
From 1995 to 2001, he served as an Aerospace Engineer in the Israeli Air
Force. His research interests include estimation theory, random process theory,
fault detection and isolation, hybrid systems, and spacecraft attitude determination.
Yaakov Oshman (A’96–SM’97) received the B.Sc. (summa cum laude) and
D.Sc. degrees, both in aeronautical engineering, from the Technion–Israel Institute of Technology, Haifa, in 1975 and 1986, respectively.
In 1987, he was a Research Associate at the State University of New York at
Buffalo, where he was, in 1988, a Visiting Professor. Since 1989, he has been
with the Department of Aerospace Engineering at the Technion, where he is
currently a Professor. During 1997–1998, he spent a sabbatical with NASA’s
Goddard Space Flight Center. His research interests are in advanced optimal
estimation, information fusion, and control methods.